Question

1. from the top of a fence, a person sites a lion on the ground at an angle of depression of 24°. if the man and the fence is 4.2 meters high, how far is the man from the lion?

Explanation:

Step1: Identify the trigonometric relationship

The angle of depression is equal to the angle of elevation (alternate interior angles). So we have a right triangle where the opposite side to the \(24^\circ\) angle is the height of the fence (\(4.2\) m) and the adjacent side is the distance \(x\) we need to find. We use the tangent function, which is \(\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}\).

Step2: Set up the equation and solve for \(x\)

We know that \(\tan(24^\circ)=\frac{4.2}{x}\). To solve for \(x\), we can rearrange the equation to \(x = \frac{4.2}{\tan(24^\circ)}\).

First, calculate \(\tan(24^\circ)\). Using a calculator, \(\tan(24^\circ)\approx0.4452\).

Then, substitute this value into the equation: \(x=\frac{4.2}{0.4452}\approx9.43\) (rounded to two decimal places).

Answer:

The man is approximately \(\boldsymbol{9.43}\) meters from the lion.